Abelian root group

If "G" is an abelian group and "P" is a set of primes then "G" is an abelian P-root group if every element in "G" has a "p"th root for every prime "p" in "P":

:$gin\; G,pin\; P\; Rightarrow\; exists\; hin\; G,\; h^p=g;$

(with the product written multiplicatively)

If the set of primes "P" has only one element "p", for convenience we can say "G" is an abelian p-root group. In a "p"-root group, the cardinality of the set of "p"th roots is the same for all elements. For any set of primes "P", being a "P"-root group is the same as being a "p"-root group for every "p" in "P".

For any specific set of primes "P", the class of abelian "P"-root groups with abelian group homomorphisms forms a full subcategory of the category of abelian groups, but not a Serre subcategory (as the quotient of an epimorphism is an abelian group, but not necessarily an abelian "P"-root group). If the set of primes "P" is empty, the category is simply the whole category of abelian groups.

If the roots are all unique, we call "G" an abelian unique "P"-root group.

If "G" is an abelian unique "P"-root group and "S" is a subset of "G", the abelian unique "P"-root subgroup generated by "S" is the smallest subgroup of "G" that contains "S" and is an abelian "P"-root group.

If "G" is an abelian unique "P"-root group generated by a set of its elements on which there are no non-trivial relations, we say "G" is a free abelian unique "P"-root group. For any particular set of primes "P", two such groups are isomorphic if the cardinality of the sets of generators is the same.

An abelian "P"-root group can be described by an abelian "P"-root group presentation:

:$langle\; g\_1,g\_2,g\_3,ldots\; |\; R\_1,\; R\_2,\; R\_3,ldots\; angle\_P$

in a similar way to those for abelian groups. However, in this case it is understood to mean a quotient of a free abelian unique "P"-root group rather than a free abelian group, which only coincides with the meaning for an abelian group presentation when the set "P" is empty.

Classification of abelian $P;$-root groups

Suppose $G;$ is an abelian $P;$-root group, for some set of prime numbers $P;$.

For each $pin\; P;$, the set $R\_p=\{gin\; G,\; exists\; nin\; mathbb\{N\},\; g^\{p^n\}=I\};$ of $p^n;$th roots of the identity as $n;$ runs over all natural numbers forms a subgroup of $G;$, called the $p;$-power torsion subgroup of $G;$ (or more loosely the $p;$-torsion subgroup of $G;$). If $G;$ is an abelian $p;$-root group, $R\_p;$ is also an abelian $p;$-root group. $G;$ may be expressed as a direct sum of these groups over the set of primes in $P;$ and an abelian unique $P;$-root group $G\_U;$:

:$G=G\_Uoplus\; left(R\_\{p\_1\}oplus\; R\_\{p\_2\}oplusldots\; ight);$

Conversely any abelian group that is a direct sum of an abelian unique $P;$-root group and a direct sum over $\{pin\; P\};$ of abelian $p;$-root groups all of whose elements have finite order is an abelian $P;$-root group.

Each abelian unique $P;$-root group $G\_U;$ is a direct sum of its torsion subgroup, $G\_T;$, all of which elements are of finite order coprime to all the elements of $P;$, and a torsion-free abelian unique $P;$-root group $G\_infty;$:

:$G\_U=G\_T\; oplus\; G\_infty;$

"G_∞" is simply the quotient of the group "G" by its torsion subgroup.

Conversely any direct sum of a group all of whose elements are of finite order coprime to all the elements of $P;$ and a torsion-free abelian unique $P;$-root group is an abelian unique $P;$-root group.

In particular, if $P;$ is the set of "all" prime numbers, $G\_U;$ must be torsion-free, so $G\_T;$ is trivial and $G\_U=G\_infty;$).

In the case where $P;$ includes all but finitely many primes, $G\_infty;$ may be expressed as a direct sum of free abelian unique $Q\_i;$-root groups for a set of sets of primes $Q\_isupseteq\; P;$.

:

In particular, when $P;$ is the set of all primes,

:

a sum of copies of the rational numbers with addition as the product.

(This result is not true when $P;$ has infinite complement in the set of all primes. If

:$forall\; iin\; mathbb\{N\},\; p\_i\; otin\; P\; ;$

is an infinite set of primes in the complement of $P;$ then the abelian unique $P;$-root group which is the quotient by its torsion subgroup of the group with the following presentation:

:$langle\; e\_1,e\_2,e\_3,ldots\; |\; e\_1=e\_2^\{p\_1\},e\_2=e\_3^\{p\_2\},e\_3=e\_4^\{p\_3\},ldots\; angle\_P$

cannot be expressed as a direct sum of free abelian unique $Q;$-root groups.)

Examples

*The angles constructible using compass and straightedge form an abelian 2-root group under addition modulo $2pi;$. Each element of this group has two 2-roots.

*The groups of numbers with a terminating decimal expansion and addition as the product is the free abelian unique $\{2,5\};$-root group with a single generator.

*The group of rational numbers with addition as the product, $\{mathbb\{Q\},+\};$, is the free abelian $P;$-root group on a single generator for $P;$ the set of all primes.

*For a prime $p;$, the group of complex numbers of the form $e^\{2pi\; i\; frac\{r\}\{p^n;$ for $r;$ and $n;$ natural numbers forms an abelian $p;$-root group $R\_p;$, all of whose elements have finite order, with the usual product. This group has a presentation as an abelian $p;$-root group:

:$langle\; g\; ;|\; g\; ;\; angle\_\{\{p;$

:This group is known as the Prüfer group, the "p"-quasicyclic group or the "p^∞" group

*The group $mathbb\{T\}^1;$ of complex numbers of modulus 1 forms an abelian $P;$-root group where $P;$ is the set of all prime numbers. $mathbb\{T\}^1;$ may be expressed as the direct sum:

:

:where each $R\_p;$ is the group defined in the previous example, $F\_Pcong\{mathbb\{Q\},+\};$, and $I;$ has the cardinality of the continuum.

ee also

*Root group